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11
An intuitionistic theory of types
"... An earlier, not yet conclusive, attempt at formulating a theory of this kind was made by Scott 1970. Also related, although less closely, are the type and logic free theories of constructions of Kreisel 1962 and 1965 and Goodman 1970. In its first version, the present theory was based on the strongl ..."
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An earlier, not yet conclusive, attempt at formulating a theory of this kind was made by Scott 1970. Also related, although less closely, are the type and logic free theories of constructions of Kreisel 1962 and 1965 and Goodman 1970. In its first version, the present theory was based on the strongly impredicative axiom that there is a type of all types whatsoever, which is at the same time a type and an object of that type. This axiom had to be abandoned, however, after it was shown to lead to a contradiction by Jean Yves Girard. I am very grateful to him for showing me his paradox. The change that it necessitated is so drastic that my theory no longer contains intuitionistic simple type theory as it originally did. Instead, its proof theoretic strength should be close to that of predicative analysis.
Typical ambiguity: trying to have your cake and eat it too
 the proceedings of the conference Russell 2001
"... Would ye both eat your cake and have your cake? ..."
Interuniversal Teichmüller Theory IV: Logvolume Computations and Settheoretic Foundations, preprint
"... The present paper forms the fourth and final paper in a series of papers concerning “interuniversal Teichmüller theory”. In the first three papers of the series, we introduced and studied the theory surrounding the logthetalattice, a highly noncommutative twodimensional diagram of “miniature mod ..."
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Cited by 4 (4 self)
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The present paper forms the fourth and final paper in a series of papers concerning “interuniversal Teichmüller theory”. In the first three papers of the series, we introduced and studied the theory surrounding the logthetalattice, a highly noncommutative twodimensional diagram of “miniature models of conventional scheme theory”, called Θ ±ellNFHodge theaters, that were associated, in the first paper of the series, to certain data, called initial Θdata. This data includes an elliptic curve EF over a number field F, together with a prime number l ≥ 5. Consideration of various properties of the logthetalattice led naturally to the establishment, in the third paper of the series, of multiradial algorithms for constructing “splitting monoids of LGPmonoids”. Here, we recall that “multiradial algorithms ” are algorithms that make sense from the point of view of an “alien arithmetic holomorphic structure”, i.e., the ring/scheme structure of a Θ ±ell NFHodge theater related to a given Θ ±ell NFHodge theater by means of a nonring/schemetheoretic horizontal arrow of the logthetalattice. In the present paper, estimates arising from these multiradial algorithms for splitting
Why sets?
 PILLARS OF COMPUTER SCIENCE: ESSAYS DEDICATED TO BORIS (BOAZ) TRAKHTENBROT ON THE OCCASION OF HIS 85TH BIRTHDAY, VOLUME 4800 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2008
"... Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a faraway planet. Would their mathematics be setbased? What are the alternatives to the settheoretic foundation of mathematics? Besi ..."
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Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a faraway planet. Would their mathematics be setbased? What are the alternatives to the settheoretic foundation of mathematics? Besides, set theory seems to play a significant role in computer science; is there a good justification for that? We discuss these and some related issues.
Enriched stratified systems for the foundations of category
"... This is the fourth in a series of intermittent papers on the foundations of category theory stretching back over more than thirtyfive years. The first three were “Settheoretical foundations of category theory ” [1969], “Categorical foundations and foundations of category theory ” [1977], and much ..."
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This is the fourth in a series of intermittent papers on the foundations of category theory stretching back over more than thirtyfive years. The first three were “Settheoretical foundations of category theory ” [1969], “Categorical foundations and foundations of category theory ” [1977], and much more recently, “Typical ambiguity: Trying to have
Three Conceptual Problems That Bug Me
, 1996
"... Introduction I will talk here about three problems that have bothered me for a number of years, during which time I have experimented with a variety of solutions and encouraged others to work on them. I have raised each of them separately both in full and in passing in various contexts, but thought ..."
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Introduction I will talk here about three problems that have bothered me for a number of years, during which time I have experimented with a variety of solutions and encouraged others to work on them. I have raised each of them separately both in full and in passing in various contexts, but thought it would be worthwhile on this occasion to bring them to your attention side by side. In this talk I will explain the problems, together with some things that have been tried in the past and some new ideas for their solution. Types of conceptual problems. A conceptual problem is not one which is formulated in precise technical terms and which calls for a definite answer. For this reason, there are no clearcut criteria for their solution, but one can bring some criteria to bear. These will vary from case to case. There are three general types of conceptual problems in mathematics of which the ones that I will discuss today are examples. These are: 1 ffi<F
Gedanken: A tool for pondering the tractability of correct program technology
, 1994
"... syntax of elementary languages in Gedanken . . . . . . . . . . . 129 7.1 Match counting algorithm for patterns over PC k . . . . . . . . . . . . . 157 8.1 log 2 speed of Model Graphs after elimination . . . . . . . . . . . . . . . 187 8.2 log 2 speedup of Model Graphs after elimination . . . . . . ..."
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syntax of elementary languages in Gedanken . . . . . . . . . . . 129 7.1 Match counting algorithm for patterns over PC k . . . . . . . . . . . . . 157 8.1 log 2 speed of Model Graphs after elimination . . . . . . . . . . . . . . . 187 8.2 log 2 speedup of Model Graphs after elimination . . . . . . . . . . . . . 188 8.3 log 2 speed of Model Graphs after invalidation . . . . . . . . . . . . . . . 188 8.4 log 2 speedup of Model Graphs after invalidation . . . . . . . . . . . . . 189 ix Chapter 1 Summary One goal of computer science has been to develop a tool T to aid a programmer in building a program P that satisfies a specification S by helping the programmer build a proof in some logic of programs L that shows that P satisfies S. S typically is a pair of propositions (#, #) such that, for an input x to P , #(x) # #(P (x)) when P is defined on x. # is called the precondition or assumption, and # is called the postcondition or assertion. The problem of finding a suitable logic L of programs and specifications and verification tool T may be generically referred to as the "FloydHoare problem", formulated around 1967 [Flo67, Hoa69]. Around 1977, Davis and Schwartz proposed an extension of the FloydHoare problem in which there are multiple assumptions and assertions, referring to the state of a program as execution passes through di#erent places # in the program [DS77, Sch77]. A placed proposition is then a pair (#, #), where # is either a line of a program or the name of a function. A placed proposition (#, #) holds when, if execution reaches # and the value of the variables X in P is V , then #(V ) is valid. A program with assumptions and assertions or praa is then a triple R = (P, E, F ) where the assumptions E and assertions F are sets of placed propositions. T...
SET THEORY FOR CATEGORY THEORY
, 810
"... Abstract. Questions of settheoretic size play an essential role in category theory, especially the distinction between sets and proper classes (or small sets and large sets). There are many different ways to formalize this, and which choice is made can have noticeable effects on what categorical co ..."
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Abstract. Questions of settheoretic size play an essential role in category theory, especially the distinction between sets and proper classes (or small sets and large sets). There are many different ways to formalize this, and which choice is made can have noticeable effects on what categorical constructions are permissible. In this expository paper we summarize and compare a number
Contents
"... Abstract. We prove a general theorem which includes most notions of “exact completion” as special cases. The theorem is that “κary exact categories ” are a reflective sub2category of “κary sites”, for any regular cardinal κ. A κary exact category is an exact category with disjoint and universal ..."
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Abstract. We prove a general theorem which includes most notions of “exact completion” as special cases. The theorem is that “κary exact categories ” are a reflective sub2category of “κary sites”, for any regular cardinal κ. A κary exact category is an exact category with disjoint and universal κsmall coproducts, and a κary site is a site whose covering sieves are generated by κsmall families and which satisfies a solutionset condition for finite limits relative to κ.
LOGVOLUME COMPUTATIONS AND SETTHEORETIC FOUNDATIONS
, 2012
"... The present paper forms the fourth and final paper in a series of papers concerning “interuniversal Teichmüller theory”. In the first three papers of the series, we introduced and studied the theory surrounding the logthetalattice, a highly noncommutative twodimensional diagram of “miniature mo ..."
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The present paper forms the fourth and final paper in a series of papers concerning “interuniversal Teichmüller theory”. In the first three papers of the series, we introduced and studied the theory surrounding the logthetalattice, a highly noncommutative twodimensional diagram of “miniature models of conventional scheme theory”, called Θ ±ellNFHodge theaters, that were associated, in the first paper of the series, to certain data, called initial Θdata. This data includes an elliptic curve EF over a number field F, together with a prime number l ≥ 5. Consideration of various properties of the logthetalattice led naturally to the establishment, in the third paper of the series, of multiradial algorithms for constructing “splitting monoids of LGPmonoids”. Here, we recall that “multiradial algorithms ” are algorithms that make sense from the point of view of an “alien arithmetic holomorphic structure”, i.e., the ring/scheme structure of a Θ ±ellNFHodge theater related to a given Θ ±ellNFHodge theater by means of a nonring/schemetheoretic horizontal arrow of the logthetalattice. In the present paper, estimates arising from these multiradial algorithms for splitting